1. For each of the following temperatures, find the equivalent temperature on the indicated scale.
a) -222.75°C on the Fahrenheit scale
_______°F
b) 98.4°F on the Celsius scale
_______°C
c) 115 K on the Fahrenheit scale
_______°F
2. The boiling point of liquid hydrogen is 20.3 K at atmospheric pressure.
a) What is the temperature on the Celsius scale?
_______°C
b) What is the temperature on the fahrenheit scale?
_______°F
3. The temperature difference between the inside and outside of a home on a cold day is 57.0°F.
a) Express this difference on the Celsius scale
_______°C
b) Express this difference on the Kelvin scale
__________ K
4. The New River Gorge bridge in West Virginia is a 518-m-long steel arch. How much will its length change between temperature extremes -16°C and 34°C?
____________ cm
5. A grandfather clock is controlled by a swinging brass pendulum that is 0.80 m long at a temperature of 21°C.
a) What is the length of the pendulum rod when the temperature drops to 0.0°C? (Round your answer to four significant figures.)
____________ mm
b) If a pendulum's period is given by T = 2π √ L/g , where L is its lenth, does the change in length of the rod cause the clock to run fast or slow?
Fast Slow Neither (Select one)
6. The density of gasoline is 7.30 x 102 kg/m3 at 0°C. Its average coefficient of volume expansion is 9.60 x 10-4 (°C)-1, and note that 1.00 gal = 0.00380 m3.
a) Calculate the mass of 11.2 gal of gas at 0°C.
____________ kg
b) If 1.000 m3 of gasoline at 0°C is warmed by 16.5°C, calculate its new volume.
_________m3
c) Using the answer to part (b), calculate the density of gasoline at 16.5°C.
________ kg/m3
d) Calculate the mass of 11.2 gal of gas at 16.5°C.
_______kg
e) How many extra kilograms of gasoline would you get if you bought 11.2 gal of gasoline at 0° rather than at 16.5°C from a pump that is not temperature compensated?
______ kg
7. The concrete sections of a certain superhighway are designed to have a length of 23.0 m. The sections are poured and cured at 10.0°C. What minimum spacing should the engineer leave between the sections to eliminate buckling if the concrete is to reach a temperature of 41.0°C? (Note: If applicable, Table 1 is available for use in solving this problem. See Textbook page 278)
_______ cm
8. One mole of oxygen gas is at a pressure of 5.60 atm and a temperature of 28.0°C.
a) If the gas is heated at constant volume until the pressure triples, what is the final temperature?
__________ °C
b) If the gas is heated so that both the pressure and volume are doubled, what is the final temperature?
__________ °C
9. An ideal gas occupies a volume of 1.2 cm3 at 20°C and atmospheric pressure.
a) Determine the number of molecules of gas in the container.
_________ molecules
b) if the pressure of the 1.2-cm3 volume is reduced to 2.0 x 10-11 Pa (an extremely good vacuum) while the temperature remains constant, how many moles of gas remain in the container?
__________ mol
10. Gas is confined in a tank at a pressure of 11.2 atm and a temperature of 26.0°C. If two-thirds of the gas is withdrawn and the temperature is raised to 74.0°C, what is the pressure of the gas reamining in the tank?
________atm
11. A weather balloon is designed to expand to a maximum radius of 24 m at its working altitude, where the air pressure is 0.030 atm and the temperature is 200 K. If the balloon is filled at atmospheric pressure and 253 K, what is its radius at liftoff?
________ m
Average Coefficient of Expansion for some Materials Near Room Temperature
Material
Average
Linear Expansion Coefficient (α)(°C)-1
Aluminum
24 x 10-6
Brass and bronze
19 x 10-6
Concrete
12 x 10-6
Copper
17 x 10-6
Glass (ordinary)
9 x 10-6
Glass (pyrex)
3.2 x 10-6
Invar (Ni-Fe alloy)
0.9 x 10-6
Lead
29 x 10-6
Steel
11 x 10-6
Acetone
1.5 x 10-4
Alcohol, ethyl
1.12 x 10-4
Benzene
1.24 x 10-4
Gasoline
9.6 x 10-4
Glycerin
4.85 x 10-4
Mercury
1.82 x 10-4
Turpentine
9.0 x 10-4
Airnote 1 at 0°C
3.67 x 10-3
Heliumnote 1
3.665 x 10-3
Note 1: “Gases do not have a specific
value for the volume expansion coefficient because the amount of expansion
depends on the type of process through which the gas is taken. The values given
here assume the gas undergoes an expansion at constant pressure (Serway and
Vuille 2012, pg 278).” (source: Serway, Raymond A., and Chris Vuille. College
Physics . 9th Hybrid. Boston, Ma: Brooks/Cole, 2012.)
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1. The British gold sovereign coin is an allow of gold and copper having a total mass of 7.988 g, and is 22-karat gold.
a) Find the mass of gold in the sovereign in kilograms using the fact that the number of karats = 24 x (mass of gold)/(total mass)
__________ kg
b) Calculate the volumes of gold and copper, repsectively, used to manufacture the coin
volume of gold _______ m3
volume of copper _______ m3
c) Calculate the density of the British sovereign coin.
________ kg/m3
2. Calculate the mass of a solid gold rectangular bar that has dimensions /wh = 5.20 cm x 14.0 cm x 29.0 cm. (The density of gold is 19.3 x 103 kg/m3
_________ kg
3. The four tires of an automobile are inflated to a gauge pressure of 2.3 x 105 Pa. Each tire has an area of 0.024 m2 in contact with the ground. Determine the weight of the automobile
_______ N
4. A 220-kg load is hung on a wire of length of 3.30 m, cross-sectional area 2.00 x 10-5 m2, and Young's modulus 8.00 x 1010 N/m2. What is its increase in length?
________ mm
5. A plank 2.00 cm thick and 16.1 cm wide is firmly attached to the railing of a ship by clamps so that the rest of the board extends 2.00 m horizontally over the sea below. A man of mass 90.0 kg is forced to stand on the very end. If the end of the board drops by 5.30 cm because of the man's weight, find the shear modulus of the wood.
_______ Pa
6. Bone has a Young's modulus of about 1.80 x 1010 Pa. Under compression, it can withstand a stress of about 1.65 x 108 Pa before breaking. Assume that a femur (thigh-bone) is 0.58 m long, and calculate the amount of compression this bone can withstand before breaking.
________ mm
7. The spring of pressure gauge shown in the figure below has a force constant of 1,080 N/m, and the piston has a radius of 1.40 cm. As the gauge is lowered into water, what change in depth causes the piston to move in by 0.750 cm?
_______ m
8.
a) Calculate the absolute pressure at the bottom of a fresh-water lake at a depth of 22.8 m. Assume the density of the water is 1.00 x 103 kg/m3 and the air above is at a pressure of 101.3 kPa.
_______ Pa
b) What force is exerted by the water on the window of an underwater vehicle at this depth if the window is circular and has a diamter of 30.6 cm?
_______ N
9. The figure below shows the essential parts of a hydraulic brake system. The area of the piston in the master cylinder is 1.8 cm2 and that of the piston in the brake cylinder is 6.4 cm2. The coefficient of friction between the shoe and wheel drum is 0.50. If the wheel has a radius of 41 cm, determine the frictional torque about the axle when a force of 48 N is exerted on the brake pedal.
______N * m
10.A small ferryboat is 4.00 m wide and 6.00 m long. When a loaded truck pulls onto it, the boat sinks an additional 4.20 cm into the river. What is the weight of the truck?
______ N
11. A spherical weather balloon is filled with hydrogen until its radius is 3.10 m. Its total mass including the instruments it carries is 13.0 kg
a) Find the bouyant force acting on the balloon, assuming the density of air is 1.29 kg/m3
b) What is the net force acting on the balloon and its instruments after the balloon is released from the ground?
_______N
c) Why does the radius of the balloon tend to increase as it rises to higher altitude
- Because the external pressure (increases/decreases) with increasing altitude (choose one)
12. The gravitational force exerted on a solid object is 5.05 N as measured when the object is suspended from a spring scale as in Figure (a). When the suspended object is submerged in water, the scale reads 3.82 N (Figure (b)). Find the density of the object.
______ kg/m3
13. Water flowing through a garden hose of diameter 2.71 cm fills a 24.0-L bucket in 1.10 min.
a) What is the speed of the water leaving the end of the hose?
______ m/s
b) A nozzle is now attached to the end of the hose. If the nozzle diameter is one-third the diameter of the hose, what is the speed of the water leaving the nozzle?
______ m/s
14. A liquid (ρ = 1.65g/cm3) flows through a horizontal pipe of varying cross section as in the figure below. In the first section, the cross-sectional area is 10.0 cm2, the flow speed is 283 cm/s, and the pressure is 1.20 x 105 Pa. In the second section, the cross-sectional area is 3.00 cm2
a) Calculate the smaller section's flow speed
______ m/s
b) Calculate the smaller section's pressure
______ Pa
15. Water moves through a constricted pipe in steady, ideal flow. At the lower point shown in the figure below, the pressure is 1.70 x 105 Pa and the pipe radius is 2.60 cm. At the higher point located at y = 2.50 m, the pressure is 1.22 x 105 Pa and the pipe radius is 1.30 cm.
a) Find the speed of flow in the lower section
______ m/s
b) Find the speed of flow in the upper section
______ m/s
c) Find the volume flow rate through the pipe.
______ m3/s
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1. The fishing pole in the figure below makes an angle of 20.0° with the horizontal. What is hte magnitude of the torque exerted by the fish about the axis perpendicular to the page and passing through the angler's hand if the fish pulls on the fishing line with a force = 103N at an angle 37.0° below the horizontal? The force is applied at a point L = 1.86 m from the angler's hands.
_________ N*m
2. Find the net torque on the wheel in the figure below about the axle through O perpendicular to the page, taking a = 7.00 cm and b = 21.0 cm (Indicate the direction with the sign of your answer. Assume that the positive direction is counterclockwise.)
________N*m
3. Calculate the net torque (magnitude and direction) on the beam in the figure below about the following axes.
a) an axis through O perpendicular to the page
magnitude ________ N*m
Direction: (Counterclockwise or Clockwise)
b). an axis through C perpendicular to the page
magnitude ________ N*m
Direction (Counterclockwise or Clockwise)
4. A uniform 33.5-kg beam of length l = 4.95 m is supported by a vertical rope located d = 1.20 m from its left end as in the figure below. The right end of the beam is supported by a vertical column.
a) Find the tension in the rope
__________ N upward
b) Find the force that the column exerts on the right end of the beam.
__________ N upward
5. A meter stick is found to balance at the 49.7-cm mark when placed on a fulcrum. When a 61.0-gram mass is attached at the 12.0 cm mark, the fulcrum must be moved to the 39.2-cm mark for balance. What is the mass of the meter stick?
__________ g
6. A 520-N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00-m-long, uniform, 140-N rod as indicated in the figure below. The left end of the rod is supported by a hinge and the right end is supported by a thin cable making a 30.0° angle with the vertical. (Assume the cable is connected to the very end of the 6.00-m-long rod, and that there are 2.00 m separating the wall from the sign.)
a) Find the (magnitude of the ) tension T in the cable.
_______ N
b) Find the horizontal and vertical components of the force exerted on the left end of the rod by the hinge. (Take up and to the right to be the positive directions. Indicate the direction with the sign of your answer).
Horizontal Component ___________ N
Vertical Component ___________ N
7. A window washer is standing on a scaffold supported by a vertical rope at each end. The scaffold weighs 204 N and is 2.9 m long. What is the tension in each rope when the 710-N worker stands 1.90 m from one end?
Smaller tension ________N
Larger tension ________N
8. Four objects are held in position at the corners of a rectangle by light rods as shown in the figure below. (The mass values are given in the table.)
m1 (kg)
m2 (kg)
m3 (kg)
m4 (kg)
2.7
1.7
3.5
1.5
a) Find the moment of inertia of the system about the x-axis
_________kg*m2
b) Find the moment of inertia of the system about the y-axis
_________kg*m2
c) Find the moment of inertia of the system about an axis through O and perpendicular to the page.
________kg*m2
9. If the system shown in the figure below is set in rotation about each of the axes listed below, find the torque that will produce an angular acceleration of 3.8 rad/2 in each case.
x axis ________ N * m
y axis ________ N * m
axis through O and perpendicular to the page ________ N * m
10. A large grinding wheel in the shape of a solid cylinder of radius 0.330 m is free to rotate on a frictionless, vertical axle. A constant tangential force of 230 N applied to its edge causes the wheel to have an angular acceleration of 0.872 rad/2.
a) What is the moment of inertia of the wheel?
_________ kg * m2
b) What is the mass of the wheel?
_________ kg
c) If the wheeel starts from rest, what is its angular velocity after 4.30 s have elapsed, assuming the force is acting during that time?
________ rad/s
11. A 245-kg merry-go-round in the shape of a uniform, solid, horizontal disk of radius 1.50 m is set in motion by wrapping a rope about the rim of the disk and pulling on the rope. What constant force would have to be exerted on the rope to bring the merry-go-round from rest to an angular speed of 0.700 rev/s in 2.00 s? (State the magnitude of the force.)
__________N
12. A horizontal 810-N merry-go-round of radius 1.70 m is started from rest by a constant horizontal force of 55 N applied tangentially to the merry-go-round. Find the kinetic energy of the merry-go-round after 2.0 s. (assume it is a solid cylinder)
___________ J
13. Four objects--a hoop, solid cylinder, a solid sphere, and a thin, spherical shell--each has a mass of 5.39 kg and a radius of 0.252 m
a) Find the moment of inertia for each object as it rotates about the axes shown in the table above.
hoop ________ kg * m2
Solid cylinder ________ kg * m2
Solid sphere ________ kg * m2
Thin, spherical shell ________ kg * m2
b) Suppose each object is rolled down a ramp. Rank the translational speed of each object from highest to lowest.
c) Rank the objects's rotational kinetic energies from highest to lowest as the objects roll down the ramp.
14. A light rod of length l = 1.00 m rotates about an axis perpendicular to its length and passing through its center as in the figure below. Two particles of masses m1 = 4.45 kg and m2 = 3.00 kg are connected to the ends of the rod.
a) Neglecting the mass of the rod, what is the system's kinetic energy when its angular speed is 2.7 rad/s?
___________ J
b) Repeat the problem, assuming the mass of the rod is taken to be 1.60 kg.
__________ J
15. A 240-N sphere 0.20 m in radius rolls without slipping 6.0 m down a ramp that is inclined at 37° with the horizontal. What is the angular speed of the sphere at the bottom of the slope if it starts from rest?
_________ rad/s
16. Each of the following objects has a radius of 0.184 m and a mass of 2.50 kg, and each rotates about an axis through its center (as in the table below) with an angular speed of 35.9 rad/s. Find the magnitude of the angular momentum of each object
a) a hoop
_______ kg * m2/s
b) a solid cylinder
__________ kg * m2/s
c) a solid sphere
__________ kg * m2/s
d) a hollow spherical shell
_______ kg * m2/s
17.
a) Calculate the angular momentum of Earth that arises from its spinning motion on its axis, treating Earth as a uniform solid sphere.
__________ J * s
b) Calculate the angular momentum of Earth that arises from its orbital motion about the Sun, treating Earth as a point particle.
18. A light rigid rod of lenth l = 1.00 m in length rotates about an axis perpendicular to its length and through its center, as shown in the figure below. Two particles of masses m1 = 4.55 kg and m2 = 3.00 kg are connected to the ends of the rod. What is the angular momentum of the system if the speed of each particle is 6.40 m/s? (Neglect the rod's mass).
19. A student sits on a rotating stool holding two 2.8-kg objects. When his arms are extended horizontally, the objects are 1.0 m from the axis of rotation and he rotates with an angular speed of 0.75 rad/s. The moment of inertia of the student plus stool is 3.0 kg * m2 and is assumed to be constant. The student then pulls in the objects horizontally to 0.28 m from the rotation axis.
a) Find the new angular speed of the student
_________ rad/s
b) Find the kinetic energy of the student before and after the objects are pulled in.
Before: __________ J
After ___________ J
BONUS QUESTION: +Dr-Mohammed Salah El-Gazzar asked me (in the comments below) A ball of mass M and radius R starts from rest at a height of 2.00 m and rolls down a 30.0° slope, as in the figure. What is the linear speed of the ball when it leaves the incline? Assume that the ball rolls without slipping.
Repeat this example for a solid cylinder of the same mass and radius as the ball and released from the same height.
BONUS ANSWER:Here is your answer:
1st Principle:
The Work-Energy theorem:
Sum of all Changes of energy is equal to work
-- If there is no friction, heat transfer, etc -- that is, if all forces acting are conservative forces, then the work will equal ZERO (0)
So:
Sum of changes in Linear kinetic Energy + Sum of changes in Rotational kinetic energy = - (negative) changes in potential energy
ΔKE(linear) + ΔKE(rotational) = ΔPE
ΔKE(linear) = (1/2)(m)(Δv^2)
2nd Principle:
For every linear variable, there is a rotational copy-cat.
- Mass, in linear kinematics is like "I" (represents moment of inertia --AKA rotational mass or rotational inertia--
- Velocity, in linear kinematics is like "ω" (Omega represents angular velocity)
ΔKE(rotational) = (1/2)(I)(Δω^2)
ω = velocity/radius
3rd Principle:
the Moment of inertia is a quantity that describes the mass as it is distributed around the axis of rotation.
So, different geometries of mass will have slightly different equations for "I"
-- In our case, we need to be able to calculate "I" for a Sphere and a Solid cylinder:
There is ONE big problem with your question at this point.
1. Is the "ball" a solid sphere (like a bowling ball) or a shell-like sphere (like a basket ball)?
I(solid sphere) = 2/5M(R^2)
I(hollow sphere) = 2/3M(R^2)
And Then finally
I(solid cylinder) = 1/2M(R^2)
So, let's do the equation for the solid cylinder, first, since we know exactly what formula to use:
--Start with the equation for the work energy theorem (shown in Eq 1, modified for all conservative forces).
Eq 1:
ΔKE(linear) + ΔKE(rotational) = -ΔPE
Eq 2/3/4
ΔKE(linear) = (1/2)(m)(Δv^2)
ΔKE(rotational) = (1/2)(I)(Δω^2)
ΔPE = mgΔh
Substitution of Eq 2/3/4 into Eq 1 gives the following:
Eq 5
(1/2)(m)(Δv^2) + (1/2)(I)(Δω^2) = -mgΔh
Convert the rotational variables into
Conversion 1
I = 1/2M(R^2)
Conversion 2
ω = v/r*
* NOTE: the use of capital "R" and lower case "r" both refer to the "radius", but convention has them being either capital or lower case in different conversions... They are the same variable, here: r = R
Subsitute conversion 1 and conversion 2 into equation 5 provides:
The mass can be divided out of this problem (all terms have mass)
Eq 7
(1/2)(Δv^2) + (1/2)(1/2(R^2))((Δv/r)^2) = -gΔh
This can be reduced via the following steps:
- multiply both sides by "2"
- Allow R^2 to cancel itself out
- divide both sides by 1.5
Eq 8
Δv^2 = (-2/1.5)gΔh
Take the square root of both sides to solve for the change in linear velocity. Since we are assuming the ball started at rest (zero velocity), then the Δv is equal to the final velocity (when it leaves the incline)
v = Square root((-2/1.5)gΔh)
-- Remember that the change in height (final height = 0, intial height was 2 meters) will be a negative number (final minus intial-- 0 - 2 = -2), so the negatives will cancel each other out.
Or
v = Square root ((-1.333)*(9.8)*(-2)) = 26.13268 m/s (FOR THE CYLINDER)
Now, assume the ball is solid sphere (not hollow) -- everything starts the same:
Eq 1:
ΔKE(linear) + ΔKE(rotational) = -ΔPE
Eq 2/3/4
ΔKE(linear) = (1/2)(m)(Δv^2)
ΔKE(rotational) = (1/2)(I)(Δω^2)
ΔPE = mgΔh
Substitution of Eq 2/3/4 into Eq 1 gives the following:
Eq 5
(1/2)(m)(Δv^2) + (1/2)(I)(Δω^2) = -mgΔh
HERE IS WHERE THINGS CHANGE -- "I" HAS A DIFFERENT CONVERSION EQUATION
Convert the rotational variables into
Conversion 1
I = 2/5M(R^2)
Conversion 2
ω = v/r*
Subsitute conversion 1 and conversion 2 into equation 5 provides:
= 103N at an angle 37.0° below the horizontal? The force is applied at a point L = 1.86 m from the angler's hands.
